MMAC with mixing

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Multiple Model Adaptive Control With Mixing Matthew Kuipers and Petros Ioannou, Fellow, IEEE

Abstract—Despite the remarkable theoretical accomplishments and successful applications of adaptive control, the field is not sufficiently mature to solve challenging control problems where strict performance and robustness guarantees are required. Critical to the design of practical control systems for these challenging applications, and currently lacking in parameter estimation-based adaptive control schemes, is an approach that explicitly accounts for robust-performance and stability specifications. Towards this goal, this paper describes a robust adaptive control approach called adaptive mixing control that makes available the full suite of powerful design tools from LTI theory, e.g., mixed- synthesis. The stability and robustness properties of adaptive mixing control are analyzed. It is shown that the mean-square regulation error is of the order of the modeling error provided the unmodeled dynamics satisfy a norm-bound condition. And when the parameter estimate converges to its true value, which is guaranteed if a persistence of excitation condition is satisfied, the adaptive closed-loop system converges exponentially fast to a closed-loop system comprising the plant and some LTI controller that satisfies the control objective. A benchmark example is presented, which is used to compare the adaptive mixing controller with other adaptive schemes. Index Terms—Multiple model adaptive control, robust adaptive control.

I. INTRODUCTION

W

HEN model uncertainties are sufficiently small, modern and linear time invariant (LTI) control theories, e.g., -synthesis [1]–[3], ensure, when possible, satisfactory closedloop objectives specified in meaningful engineering terms (frequency weights on the relevant transfer functions) are met. However, changes in operating conditions, failure or degradation of components, or unexpected changes in system dynamics may all violate the assumption of small uncertainty, particularly parametric uncertainty. The impact of such “large” uncertainty is that a single fixed LTI controller may no longer achieve satisfactory closed-loop behavior, let alone stability. What is needed is a controller that is able to monitor the plant dynamics in order to adjust its control law to compensate for such parametric uncertainty and other modeling errors [4, Sec. 1.3]. Adaptive control copes with large parametric uncertainty by tuning controller gains in response to estimated changes in the model. Since in conventional (robust) adaptive control [4], [5] the controller gains are calculated in real time based on the estimated plant model, the complicated relationship between Manuscript received May 06, 20098 December, 2009; revised November 10, 2009; accepted December 08, 2009. First published February 05, 2010; current version published July 30, 2010. This work was supported by the National Science Foundation under Grant 0510921. Recommended by Associate Editor A. Astolfi. The authors are with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90024 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2010.2042345

Fig. 1. Multiple model adaptive control architecture and adaptive mixing control supervisor (a) Multiple model adaptive control architecture: Based on observed data, the supervisor 6 selects/blends/mixes candidate controllers (b) Adaptive mixing control supervisor: The adaptive law 6 adjusts the estimate (t) to make the estimation error  (t) small. The mixing signal (t) mixes the candidate controllers by considering the estimate as the true unknown parameter.

plant parameters and and -synthesis controller gains has precluded the use of conventional adaptive versions of these modern robust compensators. By using candidate controllers designed off-line, the multiple model adaptive control (MMAC) architecture, shown in Fig. 1(a), avoids real-time controller synthesis issues and, therefore, provides an attractive framework for combining adaptive and modern robust tools. The MMAC architecture comprises called the two levels of control: (1) a low-level system multicontroller that is capable of generating finely-tuned cancalled the sudidate controls and (2) a high-level system pervisor that influences the control by adjusting the multicontroller, typically by selecting or weighting candidate controllers, based on processed plant input/output data. The focus of this paper is the presentation and analysis of a novel MMAC architecture, adaptive mixing control, that “mixes” the candidate controllers in a continuous manner based on a robust adaptive law. The multicontroller is not only capable of generating any of the candidate control laws but also, by controller interpolation, a stable mix of candidate control laws. This mixing behavior allows the multicontroller to evolve from one controller to another in a continuous manner. Moreover, provided certain conditions on the plant input are met, the adaptive mixing controller converges exponentially fast to meet the control objective. The supervisor, shown in Fig. 1(b), generates the mixing by processing the estimate of the unknown signal with a system called the mixer that plant parameters determines the level of participation of the candidate controllers. This determination is a manifestation of certainty , the candidate controllers that equivalence: at every fixed

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KUIPERS AND IOANNOU: MULTIPLE MODEL ADAPTIVE CONTROL WITH MIXING

were designed for are mixed such that closed-loop objectives are met. The MMAC concept is not new and has been around for quite some time. One recent approach is the so-called supervisory control [6]–[8], in which controller selection is made by continuously comparing in real time suitably defined norm-squared estimation errors, also referred to as performance signals, and the candidate controller associated with the smallest performance signal is placed in the loop according to an appropriate switching logic. Following the idea of supervisory control, logic-based switching and multiple models were combined with conventional adaptive control [9]–[11] with the objective of improving transient performance of conventional adaptive schemes. Also incorporating logic-based switching is the so-called unfalsified control approach [12], [13], which is a nonidentifier-based deterministic approach. The unfalsified control approach is a model free approach and differs from most other switching schemes. It relies on measured data to select the right controller. Even though the method guarantees convergence to a stabilizing robust controller, the simulation studies in [14], [15] report unacceptable transients. This is consistent with intuition: at the outset, measured input-output data may largely reflect initial conditions, resulting in the selection of a poorly performing controller for an extended period of time until measurement quality improves. Therefore the claim in [12], [13] that no knowledge of the plant model or its form is required is true for stability but not for performance unless further modifications are added as discussed in [15]. Switching-based schemes have a number of advantages. Switching in adaptive control was originally introduced as a method to overcome the loss of stabilizability in parameter estimation based adaptive control [16], [17]. Also, these schemes have the advantage of rapid adaptation to large, abrupt parameter changes. This is a desirable switching behavior. Switching, however, may exhibit undesirable behaviors that could negatively affect performance. Explained heuristically, if hysteresis is used and the true model is near the boundary of two candidate models, the supervisor may persistently select a controller that does not achieve desirable closed-loop behavior, despite observed data indicating an acceptable candidate controller is preferred. To encourage switching, the hysteresis constant may be reduced or replaced with a dwell-time logic, but at the increased risk of long-term intermittent switching between multiple controllers, resulting in transients from improper initialization of the new controller1. Another promising MMAC approach is based on the so-called robust MMAC (RMMAC) methodology that provides guidelines for designing both the candidate controller set (using mixed- synthesis tools) and the supervisor [18]–[20]. The RMMAC approach originated from the multiple model adaptive estimation/MMAC methods [21] of the 1970s, of which there have been numerous successful applications based on adaptations of these methods [22]–[24]. The RMMAC supervisor is based on a dynamic hypothesis testing scheme that generates for each candidate the posterior probability that 1State resetting schemes may reduce transients after switching. Adaptive mixing control is presented as an alternative approach that aims to avoid state-resetting due to unnecessary switching.

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its model is “closest” to the true plant. These probabilities are used to weight the candidate controller outputs, or, as done in the RMMAC/S variant, to switch into the loop the candidate controller associated with the highest posterior probability. Given accurate disturbance and noise models that satisfy the standard Kalman filter assumptions, simulations demonstrate rapid adaptation and superior performance compared to a nonadaptive mixed- compensator [20]. Acknowledged within the same reference, however, is that special care is needed to compensate for an inaccurate stochastic disturbance model. The RMMAC/XI architecture was proposed to handle a range of disturbance powers, at the cost of additional Kalman filters. Still, if the disturbance power is significantly outside the expected range, poor performance may occur. And although loss of stabilizability is not an issue (because there is no estimated model), it should be noted that no stability results have been published. The immediate motivation of adaptive mixing control is to provide an adaptive control approach that is capable of incorporating the full suite of powerful LTI tools, while avoiding some of the performance issues associated with undesirable switching phenomena and an unknown or uncertain disturbance model, offering an alternative to the existing MMAC approaches for particular applications. The unique feature of adaptive mixing control with respect to existing MMAC approaches, including RMMAC, is that the intent is not to converge to one controller, but rather a stable mix of candidate controllers. This mixing behavior avoids switching and, in turn, some of its undesirable behaviors. Also, while an adaptive mixing control scheme’s performance may be improved by incorporating a priori knowledge of the disturbance, the evaluations in Section VI and [25], where the latter focuses on a multiple estimator variant of the approach presented in this paper, demonstrate that adaptive mixing can achieve satisfactory performance despite significant perturbations in the disturbance power and bandwidth. Last, utilizing pre-computed controllers, adaptive mixing control avoids computational and existence issues of calculating controller gains when stabilizability of the estimated plant is lost.

NOTATION AND PRELIMINARIES Suppose that is a matrix. The transpose of is denoted by . For a -vector , is the Euclidean norm and the corresponding induced matrix norm of is . If is a function of time, then the norm denoted as and the truncated norm is defined of is denoted as as , where constant, provided that the integral exists. By that with , and we say that if , and consider the set Let

is a we mean exists.

for a given constant , where are some finite constants, and is independent of . We say that is -small in

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the mean square sense (m.s.s.) if . Furthermore, conand the set sider the signal

where are some finite constants. We say that is -small in the m.s.s. if . Let and be the transfer function and impulse is a response, respectively, of some LTI system. If for proper transfer function and analytic in , where denotes the real part of , then some the system norm is given by . system norm of is defined as The . The induced system norm of is given by . If and then . , denotes the boundary of the set , and If denotes the set-theoretic difference. An open ball of radius centered around point is denoted as . Throughout, denotes the standard basis vector, i.e., the component of is one; all other denotes the components are zero. The function bump function if ; otherwise, . The function is smooth and supported on . is We say that a square, bounded, piece-wise continuous satisfies exponentially stable (e.s.) if its transition matrix for some for all . be compact and be any conTheorem 1: Let . If the parameterized detectable stant in and is continuously differentiable with respect pair to , where and , then there exists a continuously differentiable matrix function , such that for all and , then the equilibrium 1) if of is e.s. for all and for some constant 2) if , then there exists a such that if the equilibrium of is e.s. . where The proof of Theorem 1 is a combination of the well-known results of [26] and the linear time varying (LTV) stability results found in [5, Theorem 3.4.11]. We would like to emphasize that the existence of the observer gain of Theorem 1 is utilized only for analysis purposes, not design. Furthermore, an explicit definition of is not required. II. A SIMPLE EXAMPLE In this section, we use a simple example to introduce the adaptive mixing control approach in a tutorial manner. Consider the uncertain plant (1)

is an unknown constant that belongs to the known where interval ; is a bounded disturbance, i.e., ; and is a multiplicative plant uncertainty. is assumed to be a proper rational transfer function that is analytic for some known . We refer to in as the nominal model. The control objective is to place the pole of the closed-loop nominal plant in the interval ; guarantee that and are bounded; and when , . guarantee and converge to zero as , where We consider control laws of the form is to be chosen such that the control objective is met for any . The question is whether a single fixed value is unknown of will meet the control objective given that . If we apply except that it belongs to the interval the above control law to the plant (1), we obtain the closed-loop system (2) , the closed-loop pole of (2) is , and When it follows from the bound that a single fixed value of cannot meet the control objective. Let us assume that the large parametric uncertainty is divided into the three smaller subintervals

called parameter subsets, so that the control objective could be met if it is known which parameter subset contains . Let us also assume that the following three fixed controllers (candidate controllers) are used based on which parameter subset contains : (3) (4) (5) If is known a priori then the control objective of placing the is met by senominal closed-loop pole in the interval lecting the appropriate candidate control law from (3)–(5) based belongs to more on which parameter subset contains . If than one parameter subset, then any of the appropriate candidate control laws may be chosen. Furthermore, overall closed-loop stability is guaranteed if

The right hand side of the above takes on its minimum value over when and . Thus, a sufficient condition for stability is . is not known a priori but is measured or estimated onIf line, a strategy is needed for constructing a control law that meets the control objective based on the real-time knowledge of . Our objective is to design a control law from the candidate control laws (3)–(5) continuous in to avoid discontinuities in the control signal that may occur as a result of noise or if the measured value of varies across parameter subsets. Discontinuities in the control signal are a practical concern

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then it follows from (8)–(10) that the one parameter subset , which has been constructed to meet the control law is control objective. What remains is to establish that the control objective is met on the parameter overlaps (6). If belongs to then the control law is of the model overlap , where , , and the form are some constants that satisfy . Thus, the closed-loop pole satisfies , and the control objective is met. A similar analysis shows that the . Therefore, control objective is also satisfied for if the conditions (8)–(10) are satisfied, the mixing control law (7) satisfies the control objective for every belonging to . such that (8)–(10) The next step is to define , consider are satisfied. For any

Fig. 2. Block diagram of adaptive mixing control scheme.

because it may lead to control chattering or poor transient performance. achieves Because the simple control law the control objective, it may not be obvious why we seek to from the candidate conconstruct the overall controller trollers and their corresponding parameter subsets . In general, the unknown parameter may not enter into the control law gains in a tractable manner (e.g., as in the case of -synthesis), making it difficult in the adaptive case to compute online the controller gains given the estimate . However, given the candidate controllers and parameter subsets, in the adaptive case) a it is straightforward to assign (or stabilizing candidate controller by testing which parameter subsets contain . and develop We follow the latter principle to construct an approach called control mixing where the candidate control laws are weighted based on the real-time information on . The parameter overlaps

(11) where

, , , and is the smooth bump function (cf. Section II). We define the mixing signal as . Now we establish whether the mixing signal is supported (11) satisfies requirements (8)–(10). Because on , is supported on ; is supported on ; and is supported on . Thus, (8)–(10) . Because the mixing signal (11) are satisfied for any satisfies conditions (8)–(10), the parameterized control law given by (7) and (11) meets the control objective. We now analyze the robustness of the mixing control scheme . (7), (11) applied to the true plant It follows from the Nyquist stability criterion that a sufficient condition for stability is:

(6) provide a domain in which the controller weights can be varied to continuously transition from one controller to another. The method for this simple example is described below. In Section V we extend the method to a general plant, and in Section VI a more complicated example involving mixed- compensators is considered. Let us consider a control law of the form

(7) , , and are stabilizing gains if , respectively, and are weights to be the conditions chosen so that for any

where

(8) (9) (10) are satisfied. We now verify that the control law (7), with (8)–(10) satisfied, guarantees that the control objective is met. If belongs to only

The right hand side takes on its minimum value over when , yielding the sufficient condition for the closed-loop stability. The implementation of the mixing control law (7) requires that is a known constant. In application, this knowledge may come by monitoring certain auxiliary signals, or it may be based on the results of an online parameter estimator, which is the approach of this paper. The parameter estimator is designed by following the procedures of [5, Sec. 2.4.1]. We rewrite the nomas , where and inal plant model . If and were available for measurement, we could , given the generate the estimation error estimate of . Because cannot be reliably measured, this definition of is not implementable. Therefore, we filter both sides of by the stable filter , where , to develop the linear parametric model (LPM)

where and are generated by filtering and . Given the , estimate , the estimation error is generated by which will be used to drive the adaptive law, whose task is to

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Fig. 3. Simulation results. (a) Plant output: Adaptive mixing control (solid); Adaptive pole placement control (dashed). (b) Estimate (t) of  = 2:5: Adaptive mixing control (solid); Adaptive pole placement control (dashed).

tune to make “small.” This LPM can be used to generate a wide-class of adaptive laws for generating the estimate of [5, Section 8.5], and we choose the adaptive law as the gradient algorithm with projection modification

otherwise

(12)

mixing control’s mild improvement in performance and robustness is because oscillations in (shown in Fig. 3(b)) caused by , , and do not affect the the modeling error terms . This is not the case in APPC; thus, control law when oscillations in caused by further excite . Because both schemes use an adaptive law with projection modification, the , which online estimates remain bounded by together with the large model uncertainty gives rise to the “sawtrajectories shown in Fig. 3(b). tooth” shaped III. GENERAL PROBLEM FORMULATION

where is the adaptive gain and is the projection opto . We refer to as the erator that restricts unnormalized estimation error to distinguish it from the nor. Completing the adaptive malized estimation error mixing control design, we combine the adaptive law (12) with by replacing with its estimate , and the adaptive mixing control scheme is shown in Fig. 2. A. Simulation For simulation purposes, we use the plant parameters , , , and . Additionally, sensor noise is simulated by substituting the noisy measurement for in the control law and parameter estimator. We use the control parameters , , , and , which were chosen by trial and error. For comparison, we also simulate an adaptive pole placement control (APPC) scheme that uses an identical parameter estimator as the adaptive mixing control scheme. The APPC scheme differs from the adaptive mixing control scheme only in the control law, where the APPC scheme replaces the mixing control law (7) with the APPC control . Moreover, the desired pole location of the APPC control law was chosen from robustness considerations. The plant output is shown in Fig. 3(a). In this example, both the adaptive mixing control and APPC schemes regulate the plant output towards a neighborhood of zero, where the adaptive mixing scheme has slightly faster convergence. Simulations ; adapshow that the APPC scheme remains stable for ; and perfect tive mixing control remains stable for remains stable for . Adaptive identification

Consider the SISO LTI plant

(13) (14) (15) where

represents the nominal plant; the vector contains the unknown parameters of ; ; is the measured value of corrupted by the bounded sensor noise , i.e., ; is an unknown multiplicative perturbation; ; The and is a bounded disturbance, i.e., control objective is to choose the plant input so that the plant output is regulated close to zero. We make the following assumptions on the plant to meet the control objective: is a monic polynomial whose degree is P1. known. . P2. Degree is proper, rational, and analytic in P3. for some known . P4. for some known compact convex set . Consider the state-space realization of (13) (16) We make the additional assumption to make control meaningful: P5. The pairs and are detectable and stabilizable, respectively, on .

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KUIPERS AND IOANNOU: MULTIPLE MODEL ADAPTIVE CONTROL WITH MIXING

It should be emphasized that both unstable and nonminimum phase plants are admissible despite requirements P1-P5. , Given is a family of candidate controllers where are rational transfer functions and denotes the index set , and the parameter partition , where each parameter subset is com. The candidate controller pact and covers , i.e., set and parameter partition has been developed such that for and each , the control law every yields a stable closed-loop system that meets some performance contains and the control is chosen as requirements. If , then a sufficient condition for stability over is (17) Let and be the boundaries of the sets and , respectively. To facilitate control mixing, the partition has an overlapand any there exist ping property: for all constants and such that . The paramthat belong to eter overlaps set is the set of all points more than one parameter subset. The candidate controllers and parameter subsets can, for example, be generated by the method of [19], modified to generate overlapping parameter subsets. Remark 1: A conventional adaptive controller could be included in the candidate controller set to account for the case . This is the topic of future work. Remark 2: As in many MMAC approaches, the complexity of the controller design increases with the number of candidate controllers, which may result in an impractical design. A priori knowledge can be used to decrease the number of unknown parameters and size of , and, in turn, decrease the number of candidate controllers. 1) The Problem: The objective of this paper is to propose a provably correct MMAC scheme which is capable of achieving 1) global boundedness of all system signals and 2) regulation of all plant signals in the absence of unmodeled dynamics, disturbances, and sensor noise. A deterministic approach is pursued because the disturbance is only known to be bounded. The unknown parameter vectors enter the plant model linearly and the plant remains detectable and stabilizable on . Therefore, in order to side-step issues associated with discontinuous switching among candidate controllers, we present a deterministic MMAC approach that tunes the multicontroller in a continuous manner based on of . the estimate IV. CONCEPTUAL FRAMEWORK The adaptive mixing control architecture comprises two systems: the multicontroller and the robust adaptive super, which in turn consists of a robust parameter estimator visor and mixer . For notational simplicity, the following defini(for some ), then is said to tions are made. If be an active parameter subset. denotes the index set of all , i.e., . active parameter subsets at as We define the set of all admissible mixing values at

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by

. The set of all admissible mixing values .

is denoted

A. Multicontroller The multicontroller , constructed from , is a dynamical system capable of generating a mix of candidate control is given by the stabilizable laws. The multicontroller and detectable state-space realization (18) is the multicontroller state vector; the where , , are of compatible dimensions; system matrices and the mixing signal and estimate is generated by the supervisor and tunes . For fixed values and , the multicontroller of has the transfer function (19) (20) satisfies three properties: The multicontroller , , and C1. The elements of are continuously differentiable in and . C2. , where and is standard basis vector. and any , C3. For all internally stabilizes the plant. Property C1 ensures that the closed-loop system varies slowly if and are tuned slowly. Property C2 allows for each candidate is a controller to be recovered. Property C3 ensures that stabilizing certainty equivalence control law for any admissible mixing signal, independent of the mixer implementation. The multicontroller can be viewed as a generalization of the multicontroller used in supervisory control [6]. Construction of the multicontroller involves interpolating the candidate controllers over the parameter overlaps . Numerous controller interpolation approaches have been proposed in the context of gain scheduling. These methods interpolate controller poles, zeros, and gains [27]; solutions of the Riccati equations design [28]; state-space coefficient matrices of balfor an anced controller realizations [29]; state and observer gains [30]; , where controller output [31]–[33], i.e., . As in gain scheduling, these interpolation methods may not satisfy the point-wise stability requirement C3 (cf. the counter examples of [34], [35]). Thus, if one of these interpolation methods is used then property C3 should be verified. Fortunately, there also exist theoretically justified methods [34]–[36], which can be used to construct the multicontroller. The following result is adapted from [34]: given Lemma 2: Consider the nominal plant by (14) and the stable coprime factorization that depends smoothly. Let the stable transfer functions , on , , and depend on smoothly and

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satisfy the double Bezout identity (omitting dependencies on and )

Suppose

, where is some compact set, and are stabilizing negative feedback controllers for any . Let each controller be given a , and we stable coprime factorization define

(21) (22) (23) for

. If the controller is given by

(24) , , are some positive constants that satisfy , then is stabilizing and if for and for . Motivated by Lemma 2, consider the multicontroller

where and

(25) (26) (27) where , , and are given by (21)–(23), respectively, and denotes a smooth projection functhe function is continuously differentiable and if tion2: . The function is used to ensure that each is only evaluated on and, therefore, stable; otherwise, may genand . erate an unbounded out-of-the-loop signal if We now examine if this multicontroller satisfies properties C1-C3. Property C1 follows immediately by inspection of the filters of Lemma 2 that define the multicontroller and because is smooth. Let us now consider property C3. and , we have for For fixed constants , and for . Thus

Fig. 4. Multicontroller implementation and robust performance formulation (a) Multicontroller structure (b) General control configuration.

Because indexes only stabilizing controllers, it follows from Lemma 2 that internally stabilizes the for some , plant, satisfying C3. Moreover, if , and C2 is satisfied. Fig. 4(a) shows an implementation of the multicontroller that avoids unbounded out-of-the-loop signals [37]. By Q-blending we mean the multicontroller scheme (25), which is given in a multiple input-multiple output (MIMO) format to emphasize that this approach is suitable for extensions of adaptive mixing control to the MIMO case. We now consider robust performance as the control objective, which is formulated in the standard general control configuration shown in Fig. 4(b), as the control objective. Motivated to work with a normalized uncertainty, we model the multiplica, where the known tive uncertainty as serves as a frequency depenstable transfer function is any stable transfer function satisfying dent weight and . The transfer matrix is assumed to conand its interconnection with the pertain the plant formance and uncertainty weights that describe the control obis the exogenous input (for example, say jective; ), and is the exogenous output. The mais formed by absorbing the trix transfer function into as shown in Fig. 4(b). We say that a concontroller troller yields robust performance if is stable , where and satisfies satisfies ; is a stable transfer matrix; and denotes . Let us assume that each yields robust performance for all . As candidate demonstrated by the following result, a Q-blending multicontroller preserves robust performance. is given by (25) Lemma 3: If the multicontroller , the candiand, for some generalized plant and date controller yields robust performance for all , and the multicontroller then for all yields robust performance with respect to . Proof: The transfer matrix has the form (cf. [2, pp. 150]), where . For any and , it follows that , for , and for . Thus

(28) 2

'

law. '

is not the same as the projection operator Pr f1g used in the adaptive can be constructed, for example, from smooth bump functions . Authorized licensed use limited to: University of Illinois. Downloaded on August 13,2010 at 17:33:11 UTC from IEEE Xplore. Restrictions apply.

KUIPERS AND IOANNOU: MULTIPLE MODEL ADAPTIVE CONTROL WITH MIXING

and,

consequently,

is stable and . Remark 3: The choice of controller interpolation scheme affects the complexity of the multicontroller. Say, for example, . An interthat the candidate controller orders are polation approach that schedules the controller gains with rewill result in a multicontroller order of . spect to The output-blending scheme results in a multi. To analyze the Q-blending scheme controller order of shown in Fig. 4(a), let us assume that the orders of and are . Then the order of the Q-blending multicontroller is . The a priori theoretical properties of the Q-blending approach should be carefully weighed against its complexity. B. Robust Adaptive Supervisor The robust adaptive supervisor, shown in Fig. 1(b), is a dynamical system that takes as input the measured plant signals and outputs the mixing signals that “config. Note that, because we are imures” the multicontroller plementing the supervisor, the multicontroller may access the . states of the supervisor, including the parameter estimate implements the mapping . The The mixer are assumed following properties of M1. is continuously differentiable. M2. Property M1, together with C1 ensures that if is tuned slowly then the closed-loop system will vary slowly. M2, together with is a certainty equivalence stabilizing C3, ensures that the controller meets controller, i.e., for any the control objective. Thus, if the control law is applied to the nominal plant (14), the closed-loop system is internally stable, and the closed-loop characteristic polynomial is Hurwitz. Furthermore, when this control law is applied to the overall plant (13), the closed-loop system is stable if (29) If dent of

satisfies C3, the design of the mixer is indepen; otherwise, one must ensure, for all , that meets the control objectives. Assuming requirement C3 is satisfied, the designer has considerable freedom in constructing , and one such approach, as in Section III and in the sequal, is to define based on the smooth bump function given in Section II. is unknown, cannot be calculated Because and, therefore, cannot be implemented. Thus, the adaptive with its estimate . The mixing control approach replaces well known counter example of Rohrs et al., [38] demonstrates that for even small modeling errors that an adaptive system may become unstable. Thus, because of the presence of mul, disturbance , and sensor noise , tiplicative uncertainty we use a robust online parameter estimator to regain as much of the robustness of the known case as possible. The robust parameter estimator comprises an error model and a robust adaptive law , also referred to as the tuner. The error model is constructed by selecting an appropriate

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parameterization of the plant model. We proceed with the design of the error model by constructing a LPM using the same technique as in Section III. The interested reader is referred to [5, Sec. 2.4.1] for a detailed description. The LPM of the nom, where inal system (14) is given by (30) (31) (32) is a design constant and is a proper stable where minimum-phase filter. The estimation error is generated by as the true parameter , i.e., regarding . We define the error model as the dynamical system and , and its output is whose inputs are the observed data . The error model is realized from (30)–(32) and has the form (33) (34) (35) where is Hurwitz and is tuned by . Note that is affine in . is connected to the true plant When the error model in the presence of the multiplicative perturbation and , bounded disturbance and noise , is given by where (36) is the modeling error term and acts as an estimation disturbance. can be chosen to mitigate the deleterious effects The filter will typically have of on estimation. For this purpose, a bandpass frequency response to filter out signal bias and disturbances at low frequencies and sensor noise and unmodeled dynamics at high frequencies. For simplicity, we choose to be analytic in . can be implemented by a wideThe robust adaptive law class of algorithms [5, Chapter 8.5] whose dynamics take the fairly general form (37) (38) (39) is the adaptive law’s state; where is an auxiliary state required by the tuning algorithm; and is the dynamic component of the normalization signal that guarantees that . We choose and to be implemented with projection modification [4], [5] to . The adaptive law is implein order to constrain mented by any of the algorithms found in [4], [5] with projection modification that guarantees , if , , or E1. E2. E3.

, if

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where is the normalized estimation error. Additionally, we assume that the robust adaptive law satisfies E4. i.e., adaptation ceases when . This assumption is satisfied by all adaptive laws of [4], [5] except those with -modification. Remark 4: If the plant parameters enter the model in a nonlinear fashion, one can often overparameterize the plant model. Overparameterization increases the class of admissible plants; thus, there may be some loss of performance, and, as in the example of [6, Sec. X], plant models that are not stabilizable may be introduced. Because the controller gains are computed off-line, adaptive mixing control avoids the computational and existence issues that arise in conventional adaptive control when stabilizability is lost. Furthermore, the adaptive law can be moddoes not remain in a specified neighborified to ensure that hood about the points that lead to a loss of stabilizability (cf. 7.6 of [5]). Remark 5: We believe that there will be no significant difficulties in extending adaptive mixing control to the MIMO case, which is currently under investigation. C. Stability and Robustness Results We now summarize the main results. Theorem 4: Let the unknown plant be given by (13) and satisfying the plant assumptions P1-P5. Consider the adaptive mixing controller with the multicontroller given by (18) and satisfying assumptions C1-C3; error model given given by (37)–(39) and by (33)–(35); robust adaptive law satisfying assumptions E1-E4; and mixer satisfying M1-M2. then as , where 1) If . Furthermore, let the multicontroller be given by the Q-blending scheme (25), and for some and let the candidate generalized plant yield robust performance for all . If controller , then as , where is the solution to (40) is the state-space realization and the triplet of some controller that yields robust performance with respect to the generalized plant . 2) There exists such that if , where and is a finite constant, then the adaptive mixing control scheme guarantees that . Furthermore, there exist constants such that (41) . where The proof is given in the Appendix . In result 1), exponential can be guaranconvergence of estimate to the true value teed provided is persistently exciting (PE). The control input can be augmented with an auxiliary signal that forces to be sufficiently rich of order , guaranteeing is PE if the plant is controllable and observable. For output tracking of controllable, observable plants, the reference signal may naturally ensure that is PE; otherwise, guaranteeing that is PE can be

Fig. 5. Mass-spring-dashpot system benchmark example (a) The two-cart system possesses an unmodeled control delay, uncertain spring constant, process disturbance, and measurement noise (b) Bode plot of nominal model k . for various values of 

=

accomplished by augmenting or to be sufficiently rich of order . The interested reader is referred to Chapters 3 and 4 of [5]. V. A BENCHMARK EXAMPLE In this section, we consider the two-cart mass-spring-dashpot (MSD) system [19] shown in Fig. 5(a). The parameters , , and are known, while is known to belong to . The is a low-frequency, stationary stochastic plant disturbance and is generated by , process that acts on is the disturbance bandwidth and the white where gaussian process has zero mean and unit intensity . The measurement is ’s displacement plus additive white , where gaussian noise, i.e., and . The control is applied to through a control channel with an maximum time delay of 0.05 s. The plant output is given by

where is the multiplicative unmodeled dyis the transfer function namics due to the time delay; ; from disturbance to plant output; ; and . is to be kept small. The performance variable In [19], an RMMAC scheme was developed. First, the candi, where each is a date controller set mixed- compensator, and the corresponding parameter parti, , , tion were generated by the and method (cf. [19].) The supervisor was constructed using the RMMAC approach. The control is generated by weighting each

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KUIPERS AND IOANNOU: MULTIPLE MODEL ADAPTIVE CONTROL WITH MIXING

controller output by the output of the supervisor, i.e., . For a fair comparison, the adaptive mixing control and supervisory control schemes utilize the candidate controller set developed in the RMMAC design. Let us now consider the design of an adaptive mixing control scheme. We start the design by enlarging each parameter subset of the multicontroller so that mixing regions are artificially introduced. After ex, by 10%, we have the panding the boundaries of , new parameter partition , , , and . The multicontroller design is completed by performing output blending, i.e., . It is clear that requirements C1 and C2 are satisfied. Using standard mixed- analysis tools, we have found property C3 holds over . Also, for comparison, an adaptive mixing control scheme was developed using the Q-blending approach to construct the multicontroller. is accomplished by The design of the mixing system , as defining the functions ,

where is the smooth bump function. The mixing signal is generated by normalizing , i.e., , and requirements M1 and M2 are satisfied. The final component of the adaptive mixing control scheme is is the robust adaptive law. The LPM used to derive the adaptive law, where (42) (43) (44) is the modeling error;

is the bandpass filter

where and . The passband , with a peak-to-peak 0.25 dB ripple; is , with a atand the stopband is tenuation level. These values were chosen to make use of the frequency range that is largely affected by , as shown graphically in Fig. 5(b), while filtering out the remaining frequencies that may become dominated by . An adaptive law using the gradient method based on integral cost is used: if if otherwise

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TABLE I MODEL ASSUMPTIONS SATISFIED

where is the estimate of and dynamic normalization signal, where

is the

The design parameters of the adaptive law were chosen by trial and error. The supervisory adaptive control scheme (labeled as SAC in Table I and the figures) is constructed by monitoring the esfor , timation error signals is chosen as the center of the interval where each constant , and and are generated by (42) and (43), respectively. Note that the filters that generate and are identical to the filters in the adaptive mixing control scheme. Thus, is experienced by both any effect attributed to the filter schemes. The supervisory scheme generates the monitoring signals , and selects the controller corresponding to the smallest monitoring signal subject to a hysteresis logic. Hysteresis logic prohibits switching unless the smallest monitoring signal is at least 0.1% smaller than the currently selected monitoring signal. The value of the hysteresis constant was chosen by trial and error based on an acceptable compromise between rapid response and reduced likelihood of switching due to noise and modeling error. Fig. 6 shows the results of two different adaptive mixing controllers: an output blending scheme and a Q-blending scheme. The output-blending scheme exhibits better transient and longterm performance compared to the Q-blending mixing scheme, which is more susceptible to bursting-type behaviors because and . the Q-blending multicontroller depends on both This makes it more susceptible to variations. Based on the above results, we chose to use the adaptive mixing control scheme with output blending to compare with the RMMAC and supervisory adaptive control schemes. Although an extensive evaluation of the adaptive mixing control,

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Fig. 6. Comparison of the plant outputs of the Q-blending and output-blending 0:325,  = 0:05, and nominal disturbance model. schemes with 

=

RMMAC, and supervisory schemes are beyond the scope of this paper3, we present some simulation results that illustrate some of the potential benefits of the adaptive mixing control scheme. and long-term Table I presents short-term output RMS values for various values of , and all model assumptions satisfied. The reand with sults of Table I are the average of five trials. While the three schemes achieve remarkable performance, the following observations were made. In general, the RMMAC takes on scheme exhibits the smallest start-up transient if values near the boundary of (see Fig. 7), whereas the adaptive mixing controller typically has the smallest if takes on values around 1 (see Fig. 8). The supervisory adaptive scheme tends to perform a number a switches before reliable data are observed, often creating a large transient (see Fig. 8). These observations are congruent with the short-term RMS values of Table I. Additionally, when takes on a value near the boundary of two candidate models, the supervisory adaptive controller may create additional transients by switching between controllers. This last observation is illustrated in Fig. 8, particularly Figs. 8(c) and 8(d). For a clear presentation, we have plotted the results pairwise, although the schemes were simulated side-by-side. Next we consider the off-nominal case in which the disturbance model’s power and bandwidth are increased to 100 and 3, respectively, and all control designs remain unchanged. The results are shown in Fig. 9. Because the increased disturbance power and bandwidth, the magnitudes of the residual signals of the RMMAC’s supervisor are larger than expected4 and causes the degraded performance5 seen in Fig. 9(a) and (b). The adaptive mixing control and supervisory adaptive schemes, however, are reasonably robust to this plant disturbance, as illustrated in Fig. 9(c) and (d). 3Due to space limitations, we have focused on the constant  case. We would like to remark that the mixing scheme has performed well for slow time-variations, say for example  (t) = 1 0:75 sin(0:01t). To cope with the faster time variation of  (t) = 1 0:75 sin(0:05t), we made the changes  = 0:4 and = 1 to achieve satisfactory performance, but at the cost of slight degradation of nominal performance. 4The residual signals become so large that numerical round-off causes divide-by-zero conditions in the RMMAC algorithm. Here, this condition is handled by outputting the supervisor’s previous output. This approach has maintained reasonable levels of the plant output in our simulations.

0

0

5RMMAC’s performance can likely be regained by extending the RMMAC/XI scheme to cope with a bounded range of a in addition to a range of 4. This comes at the cost of additional complexity.

Fig. 7. Comparison of the start-up transients of adaptive mixing control (AMC), RMMAC, and supervisory adaptive control (SAC):  = 1:75,  = 0:05, and nominal disturbance model.

VI. CONCLUSION This paper presents the adaptive mixing control approach. The motivation for adaptive mixing control is to develop a deterministic approach that is capable of achieving high-performance by utilizing well established robust LTI tools, while avoiding issues of undesirable switching behaviors and uncertain disturbance models. For the nominal case, it has been shown that the adaptive mixing control scheme drives the plant states to zero. We have also shown that when the plant parameter estimates converge to the their true values, which can be guaranteed by a persistence of excitation condition, the control objective, in terms of a LTI robust performance specification, is met exponentially. In the presence of unmodeled dynamics, noises, and disturbances, the regulation error is of the order of these modeling uncertainties in the mean square sense. The 2-cart MSD example has demonstrated that adaptive mixing control can yield satisfactory robustness to perturbations in the disturbance model, while avoiding some of the poor behaviors associated with undesirable switching. These promising results warrant further evaluations. Because discontinuous switching logic has been replaced with smooth, stable controller interpolation, adaptive mixing control schemes may not be able to respond to dramatic changes in the plant as rapidly as switching-based adaptive control schemes can. This is a disadvantage of the proposed scheme. Thus, a direction of future research is to develop adaptive control schemes that retain the desirable run-time properties of the mixing and switching schemes. The aim would be to have the supervisor switch in response to large parameter changes and then mix in response to subtle changes in the estimated model. APPENDIX Proof of Theorem 4: Let us define the parameterized conas the mixer-multicontroller intertroller connection. For any constant , has the transfer as function (20). We also define the parameterized system the plant-parameterized controller-error model interconnection, with its output chosen as . From (16), (18), (33)–(35), the parameterized system can be written compactly as

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=

Fig. 8. Simulation results with  1:02,  = 0:05, and nominal disturbance model: Adaptive mixing control (AMC), RMMAC, and supervisory adaptive control (SAC) (a) Plant output: AMC and RMMAC (b) Controller weights: AMC and RMMAC (c) Plant output: AMC and SAC (d) Controller weights: AMC and SAC.

where and the triplet is defined in the obvious manner. The closed-loop adaptive system is formed by replacing the parameter of the pagenerated rameterized system with the tuned estimates by the robust adaptive law . The closed-loop system (45) is in a form suitable for analysis using the tunability approach of [39]. Also, it has been establish in [40] that along the trathere exists a unique global solution jectories of , . , is a Step 1: Establish that for all fixed detectable pair. , Consider the adaptive law initialization where is any fixed constant in . The vectors represent the initial estimates of the coefficients of the and , respectively. Let , , and . Thus, it follows that and, because , there is no . Therefore the closed-loop system is an adaptation, i.e., , we have . Thus, it follows LTI system. Since and satisfy from (30) and (31) that the signals (46)

Similarly, the parameterized controller and also satisfy and

is an LTI system

From (46) and (47),

and

satisfy

The characteristic equation of the above system is

Since

is

a

stable minimum

phase filter and is Hurwitz by properties C3 and M2, we have that as . From the detectability of and , together with the convergence of and to zero, it follows that as . Since is a stability matrix, the convergence of and to zero implies that as . Therefore, converges to zero for because it has been shown that all when , the parameterized pair is detectable on . there Step 2: Establish that along the solutions of such that exists a function is exponentially stable. , Recall that the robust adaptive law guarantees that , . Applying [5, Lemma 3.3.2] to (36), together with and , yields (48)

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(49)

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=

Fig. 9. Simulation results with  1:02,  = 0:05, and the off-nominal disturbance model with a disturbance power of 100 1 4 and a bandwidth of 30 1 a: Adaptive mixing control (AMC), RMMAC, and supervisory adaptive control (SAC) (a) Plant output: AMC and RMMAC (b) Controller weights: AMC and RMMAC (c) Plant output: AMC and SAC (d) Controller weights: AMC and SAC.

and since that

and , it follows . Therefore, , , , where and is some constant. Because properties C1 and M1 guarantee that and , respectively, are continuously differentiable, the parameis continuously differentiable with respect terized controller is affine in and, therefore, conto . The error model tinuously differentiable. Consequently, the pair is continuously differentiable with respect to . Furthermore, and because the adaptive law guarantees that , it follows from the detectability result of Step 1 and result 2) of Theorem 1 that there exists a continuously differensuch that tiable function is e.s. provided that for some . , i.e., large or , this condition may not be For large satisfied even for small . By using the lengthy analysis approach of [5, Section 9.9.1], involving a contradiction argument, boundedness of the closed-loop signals can be proven satisfies a bound condition that is independent of provided . However, for simplicity, we continue with an alternative is chosen analysis approach, where we assume the filter is sufficiently small, say , so that for so that , the inequality is always satisfied. Therefore, is e.s., i.e., the transition matrix of satisfies for some positive constants and . Note that if , the adap, and from result 1) of Thetive law guarantees that orem 1, is e.s. Since is continuous and is compact,

, where is a slight abuse of notation and is taken . to mean Step 3: Establish boundedness and convergence of Let , where , and denotes any finite constant. Recall that is defined in (39). By applying output injection, we rewrite (45) as (50) where in Step 2 we established e.s. of the homogeneous part of (50). : By Lemma 3.3.3 of [5] and the We establish that e.s. property of , we have that (51) because is a continuous function of time. where norm to and Applying the , where is bounded (a consequence of the and the compactcontinuously differentiability of ness of ), yields (52) (53) where the second inequalities of (52) and (53) were obtained by and are subvectors of and then first recognizing that applying inequality (51). Consider the fictitious normalization . Note that because , signal and that . it follows from the definitions of

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Substituting (52), (53), and yields

into the definition of (54)

where the second inequality is obtained by using it follows that: From the definition of

.

(55) Applying the Bellman-Gronwall Lemma (cf. [5, Lemma 3.3.9]) to (55) yields the inequality , where

. Let us assume . Because , it follows that for , we have . Since , we have that , and together with (property E1) . Moreover, it follows implies that that because , which, together with . property E3, implies We now turn our attention to the injected system (50). Reis e.s., and are bounded, and is in call that . Therefore, , and, in turn, . Now we examine the mean-square properties of . From [5, Corollary and, in turn, since 3.3.3] it follows that is a subvector of . Thus, (41) holds. To summarize, the con, for some dition for stability is constants and , where is such that is e.s. the bound for . For this case, Let us consider that is a function because and (from E2 and ). Thus, since is e.s., we have , , and as . From the , it follows from (50) convergence of , and consequently as . that We now consider the case that . It follows from (16) and (18) that: that implies

is chosen such that

(56) (57) is tuned by the adaptive . From Lemma 3, the triplet is the state space realization of a controller that yields robust performance for the generalized plant . From (40) and (57), the dynamics of is given by . Because is Hurwitz and , we have . as . Therefore, where law. Let

and

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[29] M. G. Kellett, “Continuous scheduling of h-infinity controllers for a ms760 paris aircraft,” in Robust Control System Design Using H-Infinity and Related Methods, P. H. Hammond, Ed. London, U.K.: Institute of Measurement and Control, 1991, pp. 197–223. [30] R. A. Hyde and K. Glover, “The application of scheduled h-infinity controllers to a vstol airccraft,” IEEE Trans. Autom. Control, vol. 38, no. 7, pp. 1021–1039, Jul. 1993. [31] H. Buschek, “Robust autopilot design for future missile systems,” in Proc. AIAA Guidance, Navig., Control Conf., 1997, [CD ROM]. [32] J. H. Kelly and J. H. Evers, “An interpolation strategy for scheduling dynamic compensators,” in Proc. AIAA Guidance, Navigation, Control Conf., 1997, [CD ROM]. [33] H. Niemann and J. Stoustrup, “An architecture for implementation of multivariable controllers,” in Proc. Amer. Controls Conf., San Diego, CA, Jun. 1999, pp. 4029–4033. [34] D. J. Stilwell and W. J. Rugh, “Stability preserving interpolation methods for the synthesis of gain scheduled controllers,” Automatica, vol. 36, no. 5, pp. 665–671, 2000. [35] H. Niemann, J. Stoustrup, and R. B. Abrahamsen, “Switching between multivariable controllers,” Optimal Control Appl. Methods, vol. 25, pp. 51–66, 2004. [36] S. M. Shahruz and S. Behtash, “Design of controllers for linear parameter-varying systems by the gain scheduling technique,” J. Math. Anal. Appl., vol. 168, no. 1, pp. 195–217, 1992. [37] T. T. Tay, I. Mareels, and J. B. Moore, High Performance Control, ser. Systems and Control: Foundations and Applications. Boston, MA: Birkhuser, 1997. [38] C. Rohrs, L. Valavani, M. Athans, and G. Stein, “Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics,” IEEE Trans. Autom. Control, vol. AC-30, no. 9, pp. 881–889, Sep. 1985. [39] A. S. Morse, “Towards a unified theory of parameter adaptive control: Tunability,” IEEE Trans. Autom. Control, vol. 35, no. 9, pp. 1002–1012, Sep. 1990. [40] M. M. Polycarpou and P. A. Ioannou, “On the existence and uniqueness of solutions in adaptive control systems,” IEEE Trans. Autom. Control, vol. 38, no. 3, pp. 474–479, Mar. 1993.

Matthew Kuipers received the B.S. degree in electrical engineering from Northeastern University, Boston, MA, in 2003 and the M.S. and Ph.D. degrees in electrical engineering from the University of Southern California, Los Angeles, in 2005 and 2009, respectively. His research interests include robust adaptive control, multiple model approaches, nonlinear control, hypersonic aircraft modeling and control, and intellectual property law.

Petros Ioannou (S’80–M’83–SM’89–F’94) received the B.Sc. degree in mechanical engineering (with first class honors) from the University College London, London, U.K., in 1978 and the M.S. degree in mechanical engineering and Ph.D. degree in electrical engineering from the University of Illinois, Urbana, in 1980 and 1982, respectively. In 1982, he joined the Department of Electrical Engineering—Systems, University of Southern California, Los Angeles. He is currently a Professor in the same department and the Director of the Center of Advanced Transportation Technologies. He also holds a joint appointment with the Department of Aerospace and Mechanical Engineering. He is the author/coauthor of five books and over 150 research papers in the area of controls, neural networks, nonlinear dynamical systems, and intelligent transportation systems. His research interests are in the areas of adaptive control, neural networks, nonlinear systems, vehicle dynamics and control, intelligent transportation systems, and marine transportation. Dr. Ioannou received the Outstanding Transactions Paper Award from the IEEE Control Systems Society and the 1985 Presidential Young Investigator Award for his research in Adaptive Control in 1984. He has been an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the International Journal of Control, Automatica, and the IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS. He also served as a member of the Control System society on IEEE Intelligent Transportation Systems (ITS) Council Committee, and his center on advanced transportation technologies was a founding member of Intelligent Vehicle Highway System (IVHS) America, which was later renamed ITS America. He is currently an Associate Editor at Large of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and a Chairman of the International Federation of Automatic Control (IFAC) Technical Committee on Transportation Systems.

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